26th April

4th May (Tuesday)

Josha Box (Warwick)

Computing models for quotients of modular curves

In Sage or Magma, you can ask for the defining equations of modular curves of the form and it will give you the answer. For more general modular curves, however, an algorithm to determine such models did not exist until recently. I will describe such an algorithm based on the existing method for . This builds on earlier work of John Cremona

10th May

Beth Romano (Oxford)

Depth in the local Langlands correspondence

The local Langlands correspondence is a kaleidoscope of conjectures relating representations of p-adic groups, local Galois theory, and the theory of complex Lie groups. I'll discuss how the notion of depth appears on both sides of the correspondence. The positive-depth part of the correspondence becomes mysterious for small residue characteristic, and I'll talk about how a construction of Reeder--Yu uses geometric invariant theory to shed light on this area. Finally, I'll talk about my recent results that build on Reeder--Yu to give new positive-depth representations for certain exceptional groups.

17th May

Jon Chapman (Manchester)

Partition and density regularity for diagonal Diophantine systems

A system of equations is called partition regular if every finite colouring of the positive integers produces monochromatic solutions to the system. A system is called density regular if it has solutions over every set of integers with positive upper density. A classical result of Rado characterises all partition regular linear systems, whilst Szemerédi’s theorem classifies all density regular linear systems. In this talk, I will report on recent developments on these topics for non-linear systems. I will also show how techniques from analytic number theory and additive combinatorics can be used to classify partition and density regularity for sufficiently non-singular systems of diagonal polynomial equations.

Ashwin Iyengar (KCL)

The Iwasawa main conjecture and the extended eigencurve

I will explain how one can formulate the Iwasawa main conjecture for p-adic families of modular forms. I will focus specifically on a certain piece of the “extended eigencurve”, which is a mixed characteristic adic space where the p-adic families live.

Daniel Loughran (Bath)

Probabilistic Arithmetic Geometry

A theorem of Erdos-Kac states that the number of prime divisors of an integer behaves like a normal distribution (once suitably
renormalised). In this talk I shall explain a version of this result for integer points on varieties. This is joint work with Efthymios Sofos and Daniel El-Baz.

7th June

Ross Paterson (Glasgow)

Statistics for Elliptic Curves over Galois Extensions

As E varies among elliptic curves defined over the rational numbers, a theorem of Bhargava and Shankar shows that the average rank of the Mordell--Weil group E(Q) is bounded. If we now fix a number field K, is the same true of E(K)? Moreover, if K/F is a Galois extension then how does the Galois group act on E(K) "on average"? This talk will report on recent progress on these questions: answering the first in the affirmative for certain choices of K, and, after a more precise formulation, offering a partial answer to the second.

14th June

21st June

Shuntaro Yamagishi (Utrecht)

Solving polynomial equations in many variables in primes

Solving polynomial equations in primes is a fundamental problem in number theory. For example, the twin prime conjecture can be phrased as the statement that the equation $x_1 - x_2 - 2 = 0$ has infinitely many solutions in primes. Let $F \in \mathbb{Z}[x_1, \ldots, x_n]$ be a degree $d$ homogeneous form. In 2014, Cook and Magyar proved the existence of prime solutions to the equation $F(x_1, \ldots, x_n) = 0$ under certain assumptions on $F$. In particular, their result requires the number of variables $n$ to be an exponential tower in $d$. I will talk about a result related to this work of Cook and Magyar improving on the number of variables required.

28th June

11th January

18th January

Tim Browning (IST Austria)

25th January

Alex Bartel (Glasgow)

Statistics of Mordell-Weil groups as Galois modules

Let G be a finite group, let E/Q be an elliptic curve, and fix a finite-dimensional Q[G]-module V. Let F/Q run over all Galois extensions whose Galois group is isomorphic to G (together with a fixed such isomorphism for each F) and such that E(F) tensor Q is isomorphic to V as a G-module. Then what does E(F) itself look like "on average" in this family? I will report on joint work with Adam Morgan, in which we consider a particular special case of this general question. We propose a heuristics that predicts a precise answer in that case, and make some progress towards proving it. Our heuristics turns out to be an elliptic curve analogue of Stevenhagen's heuristic on the solubility of negative Pell equations.

1st February

Tom Bloom (Cambridge)

Sets without square differences

How dense can a set of integers be if no two distinct elements have a square difference? This question was first asked by Lovasz, and answered in a qualitative sense independently by Sarkozy and Furstenberg in the 1970s, who showed that such a set must have zero upper density. The best known upper bound previously came from an elaborate Fourier analytic argument due to Pintz, Steiger, and Szemeredi, in 1988. In joint work with James Maynard, we present the first improvement on this bound. Our method, although still relying on Fourier analysis and the circle method, is more direct, and the key new innovation is a new bound for the number of additive relations between rationals of small denominator, which is of independent interest, and we hope has many other applications to circle method problems.

Jan Vonk (Leiden)

Diagonal restrictions of p-adic families of modular forms

Katharina Hubner (Heidelberg)

The tame site

For a scheme of characteristic p > 0 (or mixed characteristic) etale cohomology with p-torsion coefficients does not behave very well: Smooth base change, cohomological purity, homotopy invariance, just to name a few, only hold for coefficients prime to the characteristic. The reason for this failure is the existence of wild ramification. This talk presents a modification of the etale topology that does not admit for wild ramification, called the tame site. For coefficients away from the characteristic the etale and tame cohomology groups are isomorphic and for p-torsion coefficients they are better behaved than the etale cohomology groups.

22nd February

Sally Gilles (Imperial)

Syntomic cohomology and period morphisms

In 2017, Colmez and Niziol proved a comparison theorem between arithmetic p-adic nearby cycles and syntomic cohomology sheaves. To prove it, they gave a local construction using (\phi, \Gamma)-modules theory which allows to reduce the period isomorphism to a comparison theorem between cohomologies of Lie algebras. I will explain the geometric version of this local construction and how to globalize it to get a new period isomorphism. In particular, the explicit description of this new isomorphism can be used to compare previous constructions of period morphisms and prove they are equal.

1st March

8th March

Luis Garcia (UCL)

Eisenstein classes and hyperplane complements

In recent years several authors (Sczech, Nori, Hill, Charollois-Dasgupta-Greenberg, Beilinson-Kings-Levin) have defined and studied certain group cocycles ("Eisenstein cocycles") in the cohomology of arithmetic groups. I will discuss how these constructions can be understood in terms of equivariant cohomology and characteristic classes. This point of view relates the cocycles to the theta correspondence and leads to generalisations relating the homology of arithmetic groups to algebraic objects such as meromorphic differentials on hyperplane complements. I will discuss these generalisations and an application to critical values of L-functions (joint with Nicolas Bergeron and Pierre Charollois).

15th March

Kirsti Biggs (Chalmers/Gothenburg)

5th October

Dan Fretwell (Bristol)

12th October

Valeriya Kovaleva (Oxford)

On the distribution of equivalence classes of p-adic quadratic forms

Some questions about quadratic forms can be reduced to a question about their canonical form, or equivalence class. In the statistical sense this means that one may use the distribution of equivalence classes to compute the proportion of quadratic forms with a certain property. In this talk we will show how to derive the probability that a random quadratic form over p-adics lies in an equivalence class, and give examples of applications.

19th October

Simon Myerson (Warwick)

Sifting rational points on elliptic curves

This is work in progress with Katharina Müller and Subham Bhakta. We discuss the problem of counting rational points on elliptic curves with bounded height and co-ordinates which are restricted in some way. We relate this to work of Loughran and Smeets on counting the varieties in a family which have a rational point.

26th October

Francesca Balestrieri (American University of Paris)

Strong approximation for homogeneous spaces of linear algebraic groups

Building on work by Yang Cao, we show that any homogeneous space of the form G/H with G a connected linear algebraic group over a number field k satisfies strong approximation off the infinite places with étale-Brauer obstruction, under some natural compactness assumptions when k is totally real. We also prove more refined strong approximation results for homogeneous spaces of the form G/H with G semisimple simply connected and H finite, using the theory of torsors and descent. (This latter result is somewhat related to the Inverse Galois Problem.)

Damian Rossler (Oxford)

A generalization of Beilinson's geometric height pairing

In the first section of his seminal paper on height pairings, Beilinson constructed an ℓ-adic height pairing for rational Chow groups of homologically trivial cycles of complementary codimension on smooth projective varieties over the function field of a curve over an algebraically closed field, and asked about an generalization to higher dimensional bases. In this paper we answer Beilinson's question by constructing a pairing for varieties defined over the function field of a smooth variety B over an algebraically closed field, with values in the second ℓ-adic cohomology group of B. Over C our pairing is in fact Q-valued, and in general we speculate about its geometric origin. This is joint work with Tamás Szamuely.

Adela Gherga (Warwick)

Implementing Algorithms to Compute Elliptic Curves Over

Let S be a set of rational primes and consider the set of all elliptic curves over having good reduction outside S and bounded conductor N. Currently, using modular forms, all such curves have been determined for N less than 500000, the bulk of this work being attributed to Cremona.

Early attempts to tabulate all such curves often relied on reducing the problem to one of solving a number of certain integral binary forms called Thue-Mahler equations. These are Diophantine equations of the form F(x,y) = u, where F is a given binary form of degree at least 3 and u is an S-unit.

A theorem of Bennett-Rechnitzer shows that the problem of computing all elliptic curves over of conductor N reduces to solving a number of Thue-Mahler equations. To resolve all such equations, there exists a practical method of Tzanakis-de Weger using bounds for linear forms in p-adic logarithms and various reduction techniques. In this talk, we describe our refined implementation of this method and discuss the key steps used in our algorithm.

16th November

Peter Varju (Cambridge)

The mixing time of the ax+b Markov chain

Chung, Diaconis and Graham studied the Markov chain on Z/qZ with transitions x -> 2x+B_n, where B_n is an independent sequence of random variables uniformly distributed in {-1,0,1}. They showed that the chain is approximately uniformly distributed after c_1 log_2(q) steps for almost all q, where c_1 is a constant slightly larger than 1. They asked whether it is possible to reduce the value of c_1 to 1. This was shown not to be possible by Hildebrand who showed that the chain is far from uniformly distributed after c_2 log_2(q) steps for any q, where c_2 is some constant with 1<c_2<c_1. I will talk about a joint work with Sean Eberhard, in which we determine the best possible value for the constant. This result is based on the large sieve inequality. If 2 in the definition of the chain is replaced by 10, then this argument does not work any longer, and we deduce a weaker result using Gallagher's larger sieve.

23rd November

Andreas Bode (Lyon)

30th November

Kazim Buyukboduk (Dublin)

Perrin-Riou style critical p-adic L-functions

I will report on joint work with Denis Benois, where we gave a Perrin-Riou-style construction of Bellaïche's p-adic L-function (as well as its improvements) at a $\theta$-critical point on the eigencurve. Besides the interpolation of the Beilinson-Kato elements about this point, the key input to prove the interpolative properties is a new "eigenspace-transition via differentiation" principle.

7th December

Jessica Fintzen (Cambridge)

Representations of p-adic groups and applications

The Langlands program is a far-reaching collection of conjectures that relate different areas of mathematics including number theory and representation theory. A fundamental problem on the representation theory side of the Langlands program is the construction of all (irreducible, smooth, complex) representations of p-adic groups. I will provide an overview of our understanding of the representations of p-adic groups, with an emphasis on recent progress.
I will also outline how new results about the representation theory of p-adic groups can be used to obtain congruences between arbitrary automorphic forms and automorphic forms which are supercuspidal at p, which is joint work with Sug Woo Shin. This simplifies earlier constructions of attaching Galois representations to automorphic representations, i.e. the global Langlands correspondence, for general linear groups. Moreover, our results apply to general p-adic groups and have therefore the potential to become widely applicable beyond the case of the general linear group.